Question

The differential equation of family of circles touching $x-$axis at origin is

• A. $\frac{dy}{dx}=\frac{x+y}{x^2-y^2}$
• B. $\frac{dy}{dx}=\frac{2}{x^2-y^2}$
• C. $\frac{dy}{dx}=\frac{2xy}{x^2-y^2}$
• D. $\frac{dy}{dx}=\frac{2xy}{x^2+y^2}$

Answer 1

The correct option is C $\frac{dy}{dx}=\frac{2xy}{x^2-y^2}$

Equation of family of circles touching $x-$axis at origin is
$x^2+(y-a{)}^2=a^2$ where $a$ is arbitrary constant.
$\Rightarrow x^2+y^2=2ay\cdots \left(i\right)$
Differentiating w.r.t $x$, we get
$2x+2y{y}^{\prime }=2a{y}^{\prime }$
$\Rightarrow x+y{y}^{\prime }=a{y}^{\prime }\cdots \left(ii\right)$
Divide $\left(i\right)$ and $\left(ii\right),$ we get
$\frac{x^2+y^2}{x+y{y}^{\prime }}=\frac{2y}{y^{\prime }}$
$\Rightarrow y^{\prime }{x}^2+y^{\prime }{y}^2=2xy+2y^2{y}^{\prime }$
$\Rightarrow \frac{dy}{dx}=\frac{2xy}{x^2-y^2}$